Bound states of edge dislocations

We investigate bound state solutions of the 2D Schr\"odinger equation with a dipole potential originating from the elastic effects of a single edge dislocation. These solutions are important for several physical systems, including the binding of $^3$He impurities to dislocations in solid $^4$He, and the nucleation of superfluidity on dislocations in $^4$He. We present a variational estimate for the binding energy and numerically solve the eigenvalue problem to obtain several bound states. In our nondimensionalized units, we find a ground state energy of -0.139. The energy spectrum obtained matches with that from semiclassical considerations.